Understanding the X-Intercepts and Y-Intercepts of the Equation y -2x^3 - 2x - 2

This article delves into the detailed process of finding both the x-intercepts and y-intercepts of the cubic polynomial equation y -2x^3 - 2x - 2. We'll break down the methodology and show you how to determine these intercepts step-by-step, ensuring you can confidently solve similar equations.

Introduction to the Equation

The equation in question is y -2x^3 - 2x - 2. This is a cubic polynomial, meaning it involves the variable x raised to the third power, as well as a linear term and a constant. The highest order term (-2x^3) guides us in knowing that this is a polynomial of degree 3.

Y-Intercept

The y-intercept of a polynomial is the point where the graph intersects the y-axis. At this point, x equals 0. To find the y-intercept, we substitute x 0 into the equation:

Let's start by setting x to 0 in the equation:

y -2(0)^3 - 2(0) - 2

Calculating this, we get:

y -2(0) - 2

y -2

The y-intercept is -2, and its coordinates are (0, -2).

X-Intercepts

The x-intercepts are the points where the graph of the polynomial intersects the x-axis. At these points, y equals 0. To find the x-intercepts, we set the equation to 0 and solve for x:

0 -2x^3 - 2x - 2

This equation is already factored, so we can directly identify the x-values for which y 0. The roots of the polynomial give us:

-2x^3 - 2x - 2 0

Dividing every term by -2, we get:

x^3 x 1 0

We can then solve for x:

x -3 and x 2

These values of x give us the x-intercepts of the polynomial. Therefore, the x-intercepts are (-3, 0) and (2, 0).

Conclusion

In summary, we have determined that the y-intercept of the equation y -2x^3 - 2x - 2 is -2, represented as (0, -2), and the x-intercepts are (-3, 0) and (2, 0).

Key Takeaway: By substituting x 0 into the equation to find the y-intercept and solving the equation for y 0 to find the x-intercepts, we can fully analyze and understand the behavior of the cubic polynomial equation.

Frequently Asked Questions (FAQs)

Question: What are x-intercepts and y-intercepts in a polynomial equation? Answer: X-intercepts are the points where the graph of a polynomial equation intersects the x-axis (y 0), and y-intercepts are the points where the graph intersects the y-axis (x 0). Question:

Answer: To find the y-intercept, substitute x 0 into the polynomial equation and solve for y. To find the x-intercepts, set the equation to 0 and solve for x.

Question: How can you determine the degree of a polynomial? Answer: The degree of a polynomial is determined by the highest exponent of the variable in the equation. In the equation y -2x^3 - 2x - 2, the term with the highest degree is -2x^3, which indicates a polynomial of degree 3.